Similarity Transformation
المؤلف:
Arfken, G.
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
الجزء والصفحة:
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29-9-2021
2710
Similarity Transformation
The term "similarity transformation" is used either to refer to a geometric similarity, or to a matrix transformation that results in a similarity.
A similarity transformation is a conformal mapping whose transformation matrix 
can be written in the form
 |
(1)
|
where 
and 
are called similar matrices (Golub and Van Loan 1996, p. 311). Similarity transformations transform objects in space to similar objects. Similarity transformations and the concept of self-similarity are important foundations of fractals and iterated function systems.
The determinant of the similarity transformation of a matrix is equal to the determinant of the original matrix
The determinant of a similarity transformation minus a multiple of the unit matrix is given by
If 
is an antisymmetric matrix (
) and 
is an orthogonal matrix (
), then the matrix for the similarity transformation
 |
(9)
|
is itself antisymmetric, i.e., 
. This follows using index notation for matrix multiplication, which gives
Here, equation (10) follows from the definition of matrix multiplication, (11) uses the properties of antisymmetry in 
and orthogonality in 
, (12) is a rearrangement of (11) allowed since scalar multiplication is commutative, and (13) follows again from the definition of matrix multiplication.
The similarity transformation of a subgroup 
of a group 
by a fixed element 
in 
not in 
always gives a subgroup (Arfken 1985, p. 242).
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 311, 1996.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 83-103, 1991.
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